Simplify and expand the following expression: $ \dfrac{5}{k + 4}- \dfrac{5}{5k - 45}- \dfrac{4}{k^2 - 5k - 36} $
First find a common denominator by finding the least common multiple of the denominators. Try factoring the denominators. We can factor a $5$ out of denominator in the second term: $ \dfrac{5}{5k - 45} = \dfrac{5}{5(k - 9)}$ We can factor the quadratic in the third term: $ \dfrac{4}{k^2 - 5k - 36} = \dfrac{4}{(k + 4)(k - 9)}$ Now we have: $ \dfrac{5}{k + 4}- \dfrac{5}{5(k - 9)}- \dfrac{4}{(k + 4)(k - 9)} $ The least common multiple of the denominators is: $ (k + 4)(k - 9)$ In order to get the first term over $(k + 4)(k - 9)$ , multiply by $\dfrac{5(k - 9)}{5(k - 9)}$ $ \dfrac{5}{k + 4} \times \dfrac{5(k - 9)}{5(k - 9)} = \dfrac{25(k - 9)}{(k + 4)(k - 9)} $ In order to get the second term over $(k + 4)(k - 9)$ , multiply by $\dfrac{k + 4}{k + 4}$ $ \dfrac{5}{5(k - 9)} \times \dfrac{k + 4}{k + 4} = \dfrac{5(k + 4)}{(k + 4)(k - 9)} $ In order to get the third term over $(k + 4)(k - 9)$ , multiply by $\dfrac{5}{5}$ $ \dfrac{4}{(k + 4)(k - 9)} \times \dfrac{5}{5} = \dfrac{20}{(k + 4)(k - 9)} $ Now we have: $ \dfrac{25(k - 9)}{(k + 4)(k - 9)} - \dfrac{5(k + 4)}{(k + 4)(k - 9)} - \dfrac{20}{(k + 4)(k - 9)} $ $ = \dfrac{ 25(k - 9) - 5(k + 4) - 20} {(k + 4)(k - 9)} $ Expand: $ = \dfrac{25k - 225 - 5k - 20 - 20}{5k^2 - 25k - 180} $ $ = \dfrac{20k - 265}{5k^2 - 25k - 180}$ Simplify: $ = \dfrac{4k - 53}{k^2 - 5k - 36}$